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Subdirect products of certain varieties of unary algebras
Czechoslovak Mathematical Journal, Tome 57 (2007) no. 2, pp. 573-578 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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J. Płonka in [12] noted that one could expect that the regularization ${\mathcal R}(K)$ of a variety ${K}$ of unary algebras is a subdirect product of ${K}$ and the variety ${D}$ of all discrete algebras (unary semilattices), but is not the case. The purpose of this note is to show that his expectation is fulfilled for those and only those irregular varieties ${K}$ which are contained in the generalized variety ${TDir}$ of the so-called trap-directable algebras.
J. Płonka in [12] noted that one could expect that the regularization ${\mathcal R}(K)$ of a variety ${K}$ of unary algebras is a subdirect product of ${K}$ and the variety ${D}$ of all discrete algebras (unary semilattices), but is not the case. The purpose of this note is to show that his expectation is fulfilled for those and only those irregular varieties ${K}$ which are contained in the generalized variety ${TDir}$ of the so-called trap-directable algebras.
Classification : 08A60, 08A70, 08B15, 08B26
Keywords: unary algebra; subdirect product; variety; directable algebra 
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Ćirić, M.; Petković, T.; Bogdanović, S. Subdirect products of certain varieties of unary algebras. Czechoslovak Mathematical Journal, Tome 57 (2007) no. 2, pp. 573-578. http://geodesic.mathdoc.fr/item/CMJ_2007_57_2_a3/

[1] C. J.  Ash: Pseudovarieties, generalized varieties and similarly described classes. J. Algebra 92 (1985), 104–115. | MR | Zbl

[2] S.  Bogdanović, M.  Ćirić, B.  Imreh, T.  Petković, and M.  Steinby: On local properties of unary algebras. Algebra Colloquium 10 (2003), 461–478. | MR

[3] S.  Bogdanović, M.  Ćirić, and T.  Petković: Generalized varieties of algebras. Internat. J.  Algebra Comput, Submitted.

[4] S.  Bogdanović, M.  Ćirić,  T. Petković, B.  Imreh, and M.  Steinby: Traps, cores, extensions and subdirect decompositions of unary algebras. Fundamenta Informaticae 34 (1999), 51–60. | DOI | MR

[5] S.  Bogdanović, B.  Imreh, M.  Ćirić, and T.  Petković: Directable automata and their generalizations. A survey. Novi Sad J.  Math. 29 (1999), 31–74. | MR

[6] S.  Burris, H. P.  Sankappanavar: A Course in Universal Algebra. Springer-Verlag, New York, 1981. | MR

[7] M.  Ćirić, S.  Bogdanović: Lattices of subautomata and direct sum decompositions of automata. Algebra Colloquium 6 (1999), 71–88. | MR

[8] F.  Gécseg, I.  Peák: Algebraic Theory of Automata. Akadémiai Kiadó, Budapest, 1971. | MR

[9] G.  Grätzer: Universal Algebra, 2nd ed. Springer-Verlag, New York-Heidelberg-Berlin, 1979. | MR

[10] T.  Petković, M.  Ćirić, and S.  Bogdanović: Decompositions of automata and transition semigroups. Acta Cybernetica (Szeged) 13 (1998), 385–403. | MR

[11] J.  Płonka: On the sum of a system of disjoint unary algebras corresponding to a given type. Bull. Acad. Pol. Sci., Ser. Sci. Math. 30 (1982), 305–309. | MR

[12] J. Płonka: On the lattice of varieties of unary algebras. Simon Stevin 59 (1985), 353–364. | MR